The Equivalence of Rubin's Conjecture and the Etnc/LRNC for Certain Biquadratic Extensions

Buckingham, Paul

doi:10.1017/s0017089513000281

Cited by 2 publications

(1 citation statement)

References 26 publications

(18 reference statements)

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“…Bley [5] showed that if

L

is a finite abelian extension of an imaginary quadratic field

K

and if

p

is an odd prime that splits in

K / Q

and does not divide the class number of

K

, then the ETNC for the pair

(h^{0} false(normalSpec (L) false), {double-struckZ}_{(p)} false[G false])

holds, where

Z_{false(p false)}

is the localisation of

double-struckZ

p

. Buckingham [19] considered certain relative biquadratic extensions. Now suppose

K = Q

.…”

Section: Introductionmentioning

confidence: 99%

On the p‐adic Stark conjecture at s=1 and applications

Johnston

Nickel

2020

J. London Math. Soc.

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Let E/F be a finite Galois extension of totally real number fields and let p be a prime. The 'p-adic Stark conjecture at s = 1' relates the leading terms at s = 1 of p-adic Artin L-functions to those of the complex Artin L-functions attached to E/F . We prove this conjecture unconditionally when E/Q is abelian. We also show that for certain non-abelian extensions E/F the p-adic Stark conjecture at s = 1 is implied by Leopoldt's conjecture for E at p. Moreover, we prove that for a fixed prime p, the p-adic Stark conjecture at s = 1 for E/F implies Stark's conjecture at s = 1 for E/F . This leads to a 'prime-by-prime' descent theorem for the 'equivariant Tamagawa number conjecture' (ETNC) for Tate motives at s = 1. As an application of these results, we provide strong new evidence for special cases of the ETNC for Tate motives and the closely related 'leading term conjectures' at s = 0 and s = 1.

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“…Bley [5] showed that if

L

is a finite abelian extension of an imaginary quadratic field

K

and if

p

is an odd prime that splits in

K / Q

and does not divide the class number of

K

, then the ETNC for the pair

(h^{0} false(normalSpec (L) false), {double-struckZ}_{(p)} false[G false])

holds, where

Z_{false(p false)}

is the localisation of

double-struckZ

p

. Buckingham [19] considered certain relative biquadratic extensions. Now suppose

K = Q

.…”

Section: Introductionmentioning

confidence: 99%

On the p‐adic Stark conjecture at s=1 and applications

Johnston

Nickel

2020

J. London Math. Soc.

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On the equivariant Tamagawa number conjecture for Tate motives and unconditional annihilation results

Johnston

Nickel

2015

Trans. Amer. Math. Soc.

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Abstract. Let L/K be a finite Galois extension of number fields with Galois group G. Let p be a prime and let r ≤ 0 be an integer. By examining the structure of the p-adic group ring Z p [G], we prove many new cases of the p-part of the equivariant Tamagawa number conjecture (ETNC) for the pair (h 0 (Spec(L))(r), Z[G]). The same methods can also be applied to other conjectures concerning the vanishing of certain elements in relative algebraic K-groups. We then prove a conjecture of Burns concerning the annihilation of class groups as Galois modules for a large class of interesting extensions, including cases in which the full ETNC is not known. Similarly, we construct annihilators of higher dimensional algebraic K-groups of the ring of integers in L.

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The Equivalence of Rubin's Conjecture and the Etnc/LRNC for Certain Biquadratic Extensions

Cited by 2 publications

References 26 publications

On the p‐adic Stark conjecture at s=1 and applications

On the p‐adic Stark conjecture at s=1 and applications

On the equivariant Tamagawa number conjecture for Tate motives and unconditional annihilation results

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